Rotating Object doesn't work quite right
It's actually rotating on the Z axis, but that's beside the point.
Try saving the scene, closing down Blender, then open Blender and the scene again and see if it's still like that. 
If you had a Limit Rotation constraint on the object, with the Limit X and Limit Y axes ticked, and the Min and Max values left on the defaults of zero, you would get that effect, but I think you'd have noticed that you put a constraint there. But check you haven't done that - if you're clicking around to explore, you can forget what you've done.
If the axes were simply locked, you shouldn't see a ring on the rotate widget for the locked axes, but they are visible in your animation, but if Blender has a glitch and hasn't updated the display, maybe the axes are locked.